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The Fibonacci ring class is a unit but its simple object is not tensor-invertible

Statement

In the coherent Fibonacci category Fibonacci fusion data and chiral convention, with simple objects $1, τ$ and $τ \otimes τ ≅ 1 \oplus τ$ , the fusion-ring class $[τ]$ is a unit but the object $τ$ is not tensor-invertible in the sense of Tensor-invertible objects, fusion-ring units, and invertible operators.

Facts & Assumptions

Given: The complex finite semisimple Fibonacci category and its simple fusion rule.

[F1]

Fibonacci fusion data and chiral convention: the simple labels are $1, τ$ and $τ \otimes τ = 1 \oplus τ$ .

[F2]

Fusion rules: simple-object multiplicities are nonnegative integers and form the basis coefficients in $K_{0} (C)$ .

[F3]

Tensor-invertible objects, fusion-ring units, and invertible operators: tensor-invertibility requires an object right inverse; an object has a finite semisimple decomposition with nonnegative multiplicities.

Proof

Given: The Fibonacci fusion rule and the definitions in [F2]–[F3].

1.1F1F2algebra

In $K_{0} (C)$ , [F1] gives $[τ]^{2} = [1] + [τ]$ . Hence $[τ] ([τ] - [1]) = [1]$ and $([τ] - [1]) [τ] = [1]$ by the same polynomial calculation. Thus $[τ]$ is a fusion-ring unit, with inverse the virtual class $[τ] - [1]$ .

2.1F1F2F3step 1.1algebra∎

If an object $Y$ were a right tensor inverse, [F3] would give $[Y] = m [1] + n [τ]$ for some integers $m, n \geq 0$ . Then $[τ] [Y] = n [1] + (m + n) [τ]$ . Equality with $[1]$ would force $n = 1$ and $m + n = 0$ , impossible for $m, n \geq 0$ . Therefore $τ$ has no right tensor inverse and cannot be tensor-invertible.

Depends on

Used by

Fibonacci path projectors and categorical noninvertibility Example

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Sources

David Penneys, Unitary Fusion Categories, Math 8800 lecture notes (standard reference, not scraped)
Shu-Heng Shao, What’s Done Cannot Be Undone: TASI Lectures on Non-Invertible Symmetries (standard reference, not scraped)